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Allocation of Risk Through Mudarabah

In this section we wish to establish the first order condition for determining an optimal risk allocation or risk pricing between the mudarabah partners. In order to establish this first order condition the following four assumptions are made:

For each contractual joint venture under mudarabah an optimal level of total profit, set at the normal profit level, is given. The mudarabah partners undertake to share this total profit in a given desired way, depending upon the postulate of risk-return criterion formulated in the foregoing section and formalised below.

For a given level of total profit through mudarabah, the corresponding level of risk is also determinate and given. Like profit-sharing, the total risk is also jointly shared under mudarabah in due percentage shares.

When the total profit and risk are optimally allocated among the mudarabah partners, there would not be any more external benefits through allocative efficiency. The mudarabah partners have given their best to each other before arriving at the optimal risk-sharing situation.

Externality associated with X-efficiency would exist in an optimal risk allocation situation. This X-efficiency arises due to non-market forces generated through the altruistic concern for mutual welfare.

Now let,

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U1 =U1 (nvir2, o2^, o22), denote a general form of a utility function for the firm,

U2 = U2 o,, a 2), denote a general form of a utility function

for the public corporation,

Where TT

Denotes expected profit for the firm,

Denotes expected profit for the public corporation,

Denotes the risk for the firm and by standard practice, it is the variance of stochastic returns from an uncertain investment by the firm,

Denotes the risk for the firm and by standard practice, it is the variance of stochastic returns from the same uncertain investment by the public corporation.

Let,

o2 denotes the total risk in a joint investment venture, i.e.

a2 = o2} + o22 +2raj a2, r being the correlation coefficient between risky returns accruing to the two partners,

77 denotes the total profit from a joint investment, i.e.

77 = 77j + 7tj.

In order to establish an optimal trade-off in risk-bearing by the partners, we formulate the optimization problem,

Max U* =U, (7t,,7r2,a21,o22)+ X[ U2 (7^, 7r2,a2 ,o22)- U*]

given, tt = 7r1 +772

a2 = o2^ + o22 +2r0^ a

and where, X is the Lagrangian multiplier.

Note that in the form that the utility functions are expressed they appear as interdependent utility functions. This is in accordance with assumption (iv) given above.

Assumption (iii) implies that,

3U2/97r1 = 0

3U2/do21 =0

aiya^ =o au1/aa22 =o.

It can then be shown as proved in the Technical Appendix at the end of this paper that,f

au1/a7r1 au1/aa21

au2/37r2 a. au2/ao22

where, A is a linear function of the regression coefficient of the risky returns of the firm on the risky returns for the corporation.

Further, let,

3U1 /3a21 = Pa1

3U2/aa22 = Pa2

Pa.j, Pa2 can be interpreted as the profit-sharing rates, and in the optimal risk-bearing situation, they also represent the price for riskbearing by the firm and the corporation, respectively, rewarded exactly by the return of profit-sharing rates.

The first order risk-bearing optimality condition under mudarabah now reduces to,

au1/37r1 au2/37r2

Pa1 A. Pa2

meaning, that the marginal utilities of a return to the firm and to the corporation are proportional to their profit-sharing rates, respectively. This was revised in the light of the comments of Dr. Asghar Qadir (Editors).

Now, by assumption (iv) as mentioned above, X-efficiency generated by mudarabah creates higher efficiency for the cooperating firm, higher profits through increase in inter-plant and intra-plant motivational efficiency, and thereby, higher confidence on the firm by the public corporation. This increases 3U2/d7r2, as indicated by the first order condition. Thus, the interdependence of U1 and U2 again.

In general, Pa, ^ Pa2, depending upon the magnitude of risk-sharing by the firm and the corporation. Let,

Pa, = (1 + P)Pa2 where, P denotes a suitable percentage. Then,

3U,/3tt, 1+P

3U2/37r2 A

The percentage, P, clearly denotes a percentage share of profit rate by the firm in the above formulation. In the optimal risk pricing situation this percentage share of the profit rate would equally demand a percentage share of risk. Hence, P gives a source for the risk component of the capitalization rate. However, it is only a part of P, say aP that enters the capitalization rate to account for the risk-sharing in a mudarabah partnership. 0 <a < 1, is possibly a percentage cost of investment.

In this section we have shown that in risky capital projects the risk component of the capitalization rate is determined by the percentage share of risk by the firm in a mudarabah partnership. This percentage share of risk is that which is determined by an optimal risk allocation condition in a mudarabah partnership.

The other component of the capitalization rate for valuation models in an Islamic economy is a required rate of return. That is, it is a rate which must necessarily exist for discounting both uncertain as well as riskless projects. We now turn to an analysis of this component.